1. Introduction

In this project, possible associated variables with the number of sold houses between the \(1^{st}\) of January 2007 and the \(30^{th}\) of June 2016 on the Stirling Ackroyd real estate company were analyzed.

In this report the index Chinese Yuan to Sterling will be explored and investigated. The yuan is the base unit of a number of former and present-day Chinese currencies, and usually refers to the primary unit of account of the renminbi, the currency of the People’s Republic of China. Having used decimal units for at least 2000 years, the yuan was probably the first currency decimal currency system. It is also considered the first to use metal coins and bank notes.

2. Data Manipulation

The data used in this report were taken from Quandl Fincancial and Economic Data. The file has 2 columns and 137 observations. As described above, the original data were manipulated and reduced to 114 observations (to match the time of interest) and a column containing only the year of the observation was added.

3. Analysis

The dataset were summarized and some results can be seen below:

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   8.925   9.782  10.220  10.900  10.990  15.520

The Box Plot is a exploratory graphic used to show the distribution of a dataset. In the present dataset the biggest value is 15.5203 , 25% of data is greater than 11.0098, 50% of data is greater than 10.22405 (median value) and the smallest value is 8.9249. The standard deviation is 1.8296867 and the amplitude is 6.5954. It is of interest to analise how the variable behaves along time, as it can be seen in the plot below.

Between 2008 and 2009 it’s possible to observe the global crisis effects in the exchange of the currencies. After the crisis, the behavior of the series varies a lot but not the value of exchange which is an indication that the variable is not a good predictor of the Number of Sold Houses.

In the analysis of time series it is common that the observations are correlated among time,this characteristic is called autocorrelation. The Durbin-Watson test is a popular option to check the hypothesis of autocorrelation. In a confidence level of 95% the output of the test is a value (p-value) between 0 and 1: if (p-value \(>\) 0,05) the null hypothesis of non-autocorrelation is not rejected, otherwise (p-valor \(\leq\) 0.05) we assume that the observations of the series are correlated among time. Also, the autocorrelation function (ACF) tests the significance of the coefficient among time (lags). The p-value is smaller than 0,05, so we assume that autocorrelation is significant.

## 
##  Durbin-Watson test
## 
## data:  base_index ~ base_date
## DW = 0.065903, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

Something also very important in the analysis of time series is to check if the series has a stationary behavior or not. The test used in this report is the Dickey-Fuller test. Stationarity is one way of modeling the dependence structure. It turns out that a lot of nice results which holds for independent random variables holds for stationary random variables. And of course it turns out that a lot of data can be considered stationary, so the concept of stationarity is very important in modeling non-independent data.

The Dickey-Fuller test for this case presents p-value of 0.2858463. So, in a 5% level of significance, the series is non-stationary

To observe and compare the variation among the years of the Chinese Yuan to Sterling the plot below shows a BoxPlot for the index in each year separately.

As expected, in time of crisis the exchange rates tend to vary a lot while during times of stability the tendency is that the exchange rate do not change considerably in short periods of time. This can be clearly seen comparing the boxes of 2008 and 2009 with the others.

As explained in section 1, the aim of the report is to verify the relationship between the Chinese Yuan to Sterling index and the Number of Sold Houses on the Stirling Ackroyd real estate company. The first exploratory analysis is to check the Pearson Correlation between the two variables, as shown below.

## 
##  Pearson's product-moment correlation
## 
## data:  base_index and house.sold$numSoldHouses
## t = 10.189, df = 112, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.5841618 0.7782118
## sample estimates:
##      cor 
## 0.693563

It can be seen that the biggest values are very influential and they are from the years of 2007 and 2008.

In the output there are two main informations: the correlation coefficient (0.693563) and the p-value (1.200171410^{-17}). The correlation coefficient (CC) can be interpreted as a measure of the degree of linear relationship between two variables and the p-value is a test to check the CC is significant: if (p-value \(>\) 0,05) the null hypothesis of correlation equals to 0 is not rejected, otherwise (p-valor \(\leq\) 0.05) the correlation is significant. In this case, the linear relationship between the two variables is positive and moderate. As a complementary analysis, the plot shown above is a scatter plot between the variables and a regression line.

A point that has to be considered is that the data from the two variables are from quite a long time (9 years) and economic changes can be notice in short periods of time. Taking this into account the correlation coefficient will be analized considering different periods of time: i) the last 5 years and ii) the last 3 years. The output from the correlation test and coefficient can be seen below and right beneath that a scatter plot is displayed, aiming to observe the existence of linear relationship.

i) Last 5 years

## 
##  Pearson's product-moment correlation
## 
## data:  base.5[, names(base.5) == variavel[2]] and house.sold.5$numSoldHouses
## t = -2.046, df = 58, p-value = 0.0453
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.481643637 -0.005914964
## sample estimates:
##      cor 
## -0.25945

The correlation coefficient is significant , in a 95% confidence level, when only the 5 last years are considered (p-value = 0.0453021), so there is significant linear relationship between the two variables.

The linear relationship is negative and weak. Also, the observed p-value is very close to the non-rejection area (p-value >0.05) and the 95% confidence interval almost includes the 0 value (which means no significant CC). In short, the linear relationship can be disregarded in this case.

ii) Last 3 years

## 
##  Pearson's product-moment correlation
## 
## data:  base.3[, names(base.3) == variavel[2]] and house.sold.3$numSoldHouses
## t = -2.3059, df = 34, p-value = 0.02734
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.62122768 -0.04460036
## sample estimates:
##        cor 
## -0.3677476

Using only the last 3 years, the correlation coefficient is significant and the linear relationship between the variables is negative and weak. The linear relationship becomes stronger when only the 3 last years are included, although is still not very useful.